§ Simply connected spaces

§ Simply connected => all paths between two points are homotopic.

If x,yx, y are two points, then there is a single unique homotopy class of points from xx to yy. Consider two paths from xx to yy called α,β\alpha, \beta. Since β1απ1(x,x)=1\beta^{-1} \circ \alpha \in \pi_1(x, x) = 1, we have that β1αϵx\beta^{-1} \circ \alpha \simeq \epsilon_x. [ie, path is homotopic to trivial path ]. compose by β\beta on the left: This becomes αβ\alpha \simeq \beta.
  • This is pretty cool to be, because it shows that a simply connected space is forced to be path connected. Moreover, we can imagine a simply connected space as one we can "continuously crush into a single point".